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denumerable infinity, as first stated: the substitutions can be
counted off 1,2,3, ... , and are not as numerous as the points ''
on a line. Poincare actually constructed such functions and
developed their most important properties in  a  series of
papers in the 1880*8. Such functions are.called automorphic.
Only two remarks need be made here to indicate what Poin-
care achieved by this wonderful creation. First, his theory
includes that of the elliptic functions as a detail. Second, as the
distinguished French mathematician Georges Humbert said,
Poincare found two memorable propositions which 'gave him
the keys of the algebraic cosmos':
Two automorphic functions* invariant under the same group
are connected by an algebraic equation;
Conversely, the co-ordinates of a point on any algebraic
curve can be expressed in terms of automorphic functions, and
hence by uniform functions of a single parameter (variable).
An algebraic curve is one whose equation is of the type
P(x,y) = 0, where P(x,y) is a polynomial in x and y. As a simple
example, the equation of the circle whose centre is at the origin
- (0,0) - and whose radius is 0, is #2 + y2 = a2. According to the
second of Poincare's *keys% it must be possible to express a?,y as
automorphic functions of a single parameter, say t. It is; for if
x = a cos t and y == a sin , then, squaring and adding, we get
rid of t (since cos21 + sin21 = 1), and find #2 + yz = a2. But the
trigonometric functions cos t, sin t are special cases of elliptic
functions, which in turn are special cases of automorphic
The creation of this vast theory of automorphic functions was
but one of many astonishing things in analysis which Poinear6
did before he was thirty. Nor was all his time devoted to analy-
sis; the theory of numbers, parts of algebra, and mathematical
astronomy also shared Ms attention. In the first he recast the
Gaussian theory of binary quadratic forms (see chapter on
* PoincarS called some of his functions 'Fuchsian*, after the
German mathematician Lazarus Fuchs (1833-1902) one of the
creators of the modern theory of differential equations, for reasons
that need not be gone into here. Others he called 'Kleinian' after
Felix Klein - in ironic acknowledgement of disputed priority.